When you refinance, you are going into a new loan. Depending on the specifics of that new loan, you can be in another 30 year loan, a 15 year loan, or some other type of loan product. Interest rates are really low so I'd compare what you are paying now to what you could be paying and if you could pay more, then you could pay your loan off faster. However, I don't think I'm the best person to give you advice, I'd speak with a loan officer.
You can always use a mortgage calculator to figure out where you stand and where you will be before speaking with a loan officer. I think this is a pretty good calculator. http://alliancedirecthm.com/19-mortgage-calculators/
Absolutely yes Refinancing to lower your monthly payment is great - unless it hurts you significantly in the long run. If you're several years into a 30-year mortgage, you've paid a lot of interest but not much principal. Refinancing into a 15-year mortgage will probably increase your monthly payment, possibly to a level that you can't afford. If you start over again with a new 30-year mortgage, you're starting with almost as much principal as you had at the beginning of your current mortgage. While your new interest rate will be lower, you'll be paying it for 30 years. So your long-term savings might be insignificant or the loan might even cost you more in the long run
Wanted to learn the hardship letter fro your mortgage communication especially in dealing with your loan modifications and foreclosed property.
Hope this a help!